In the field $No$ of surreal numbers, does $\underbrace {\frac 1 \omega + \frac 1 \omega + ...}_{\omega\text{ times}}= 1?$

Question

The question is in the title. It is known that $\omega$ $\cdot$ $\frac 1 {\omega}$ = 1, but can the expression on the left-hand side be replaced by the infinite sum in the title? If so then by the fact that $No$ is a field suggests that $\frac 1 {\omega}$ can be thought of as a probability measure. On the other hand, consider the following examples given by Prof. Conway (onpp.43-44 of On Numbers and Games:

It is interesting to note that our definitions of infinite sums have in a certain sense to be "global", rather than as limits of partial sums, because limits don't seem to work. For instance, the limit of the sequence 0, $\frac 1 2$, $\frac 2 3$, $\frac 3 4$,... ($\omega$ terms) is not 1, at least in the ordinary sense, because there are plenty of numbers in between. A simpler, but sometimes less convincing, example of the same phenomenon is given by the sequence

0,1,2,3,...

of all finite ordinals, which one would expect to tend to $\omega$, but obviously can't, since there is a whole Host of numbers greater than every finite integer but less than $\omega [here Prof. Conway gives us his favorites of such numbers--my comment].

Answer 1

You seem to be claiming that there is an obviously-natural definition of infinite sums of surreals that should be applied to make sense of $\dfrac{1}{\omega}+\dfrac{1}{\omega}+\cdots$. But:

1. The definition of "more general infinite sums" given at the top of page 40 of On Numbers and Games (ONAG) cannot apply because for the relevant $y$, we would have $\sum_nr_{n,y}=\sum1$ which does not converge.
2. The point of the section you quote is basically "there isn't a natural definition of limit to use here", so it's not clear there's a definition of limit you can lean on to avoid the problem in #1.

But if it helps, we can note that $1$ is the simplest number greater than each of $\dfrac{n}{\omega}$ for naturals $n$.